Amplification and quantum statistics of microcavity polaritons under nonresonant excitation

A model describing amplification and quantum statistics of the exciton polaritons with $\mathbf{k}=0$ in a nonresonantly excited semiconductor quantum well embedded in a microcavity is presented. Exploiting the bottleneck effect for exciton polaritons, it is assumed that the polaritons with $\mathbf{k}\ne 0$ act as a reservoir. The time evolution of the $\mathbf{k}=0$ polaritons is described by a master equation, in which two- and one-polariton transitions between the mode with $\mathbf{k}=0$ and the reservoir are accounted for. The $\mathbf{k}=0$ mode exhibits a threshold depending on the material parameters and on the injected exciton density. Below threshold, the quantum statistics of the emission is characteristic of an incoherent process, while above threshold it approaches that of a laser.

Figure 1

A sketch of the nonresonant excitation, the exciton formation and relaxation to $k=0$ in a quantum well embedded in a microcavity.

Figure 2

Gain and amplification of the mode $k=0$ as a function of the polariton density in the bottleneck, i.e., of the strength of the external pump for a CdTe quantum well.

Figure 3

Plot of the polariton distribution $\rho \left(n\right)$ for a CdTe quantum well for different polariton densities in the bottleneck.

Figure 4

Plot of first moment of the polariton distribution $\rho \left(n\right)$ for a CdTe quantum well as a function of the exciton density in the bottleneck. It clearly shows amplification at threshold. For comparison, the same first moment is presented for a one-mode laser.

Figure 5

Plot of the second normalized moment $M_2$ of the polariton distribution $\rho \left(n\right)$ for a CdTe quantum well as a function of the exciton density in the bottleneck. For comparison, the same second normalized moment is presented for a one-mode laser.

Figure 6

Plot of the linewidth of the polariton emission for a CdTe quantum well as a function of the exciton density in the bottleneck. The linewidth decreases and attains its minimum at threshold.

Figure 7

Plot of the polariton amplitude for a CdTe quantum well as a function of the exciton density in the bottleneck. For comparison, the mean field approximation is also indicated.